where \(A(r)=(r^2+m^2-g^2\vert B_E\vert ^2)/(-2igr\vert B_E\vert )\) .
Both the angular and the resulting radial integrals are standard [1]}.
In terms of the rescaled quantities \(\tilde{m}=m\Lambda ^{-1}\)  and
\(\tilde{g}=g \vert B_E\vert \Lambda ^{-1}\) ,
the latter of which is proportional to the amplitude of the vector background field, (REF ) becomes:
\(\begin{split}I_{aPT_1}=\frac{i \Lambda ^3 \pi ^2 \tilde{m}}{\tilde{g}^2}\Bigl \lbrace &1+(\tilde{m}^2-\tilde{g}^2)(2+\tilde{m}^2+\tilde{g}^2)-(1+\tilde{m}^2+\tilde{g}^2 )\sqrt{(1 + \tilde{m}^2 + \tilde{g}^2)^2 - 4 \tilde{g}^2 \tilde{m}^2}\\&+4\tilde{m}^2\tilde{g}^2 \ln \Big [ \frac{1}{2 \tilde{m}^2}\Bigl ( \sqrt{(1 + \tilde{m}^2 + \tilde{g}^2)^2 - 4 \tilde{g}^2 \tilde{m}^2}+ 1 + \tilde{m}^2 + \tilde{g}^2 \Bigr ) \Big ]\Bigr \rbrace  .\end{split}\) 
